Matt Holzer

Criteria for pointwise growth and their role in invasion processes

This article is concerned with pointwise growth and spreading speeds in systems of parabolic partial differential equations. Several criteria exist for quantifying pointwise growth rates. These include the location in the complex plane of singularities of the pointwise Green’s function and pinched double roots of the dispersion relation. The primary aim of this work is to establish some rigorous properties related to these criteria and the relationships between them. In the process, we discover that these concepts are not equivalent and point to some interesting consequences for nonlinear front invasion problems. Among the more striking is the fact that pointwise growth does not depend continuously on system parameters. Other results include a determination of the circumstances under which pointwise growth on the real line implies pointwise growth on a semi-infinite interval. As a final application, we consider invasion fronts in an infinite cylinder and show that the linear prediction always favors the formation of stripes in the leading edge.

A slow pushed front in a Lotka-Volterra competition model

We study invasion speeds in the Lotka–Volterra competition model when the rate of diffusion of one species is small. Our main result is the construction of the selected front and a rigorous asymptotic approximation of its propagation speed, valid to second order. We use techniques from geometric singular perturbation theory and geometric desingularization. The main challenge arises from the slow passage through a saddle-node bifurcation. From a perspective of linear versus nonlinear speed selection, this front provides an interesting example as the propagation speed is slower than the linear spreading speed. However, our front shares many characteristics with pushed fronts that arise when the influence of nonlinearity leads to faster than linear speeds of propagation. We show that this is a result of the linear spreading speed arising as a simple pole of the resolvent instead of as a branch pole. Using the pointwise Greens function, we show that this pole poses no a priori obstacle to marginal stability of the nonlinear travelling front, thus explaining how nonlinear systems can exhibit slower spreading that their linearization in a robust fashion.

Anomalous spreading in a system of coupled Fisher-KPP equations

This article is concerned with the rigorous validation of anomalous spreading speeds in a system of coupled Fisher-KPP equations of cooperative type. Anomalous spreading refers to a scenario wherein the coupling of two equations leads to faster spreading speeds in one of the components. The existence of these spreading speeds can be predicted from the linearization about the unstable state. We prove that initial data consisting of compactly supported perturbations of Heaviside step functions spreads asymptotically with the anomalous speed. The proof makes use of a comparison principle and the explicit construction of sub and super solutions.

Accelerated Fronts in a Two-Stage Invasion Process

We study wavespeed selection in a staged invasion process. We consider a model in which an unstable homogeneous state is replaced via an invading front with a secondary state. This secondary state is also unstable and, in turn, replaced by a stable homogeneous state via a secondary invasion front. We are interested in the selected wavespeed of the secondary front. In particular, we investigate conditions under which the influence of the primary front increases this speed. We find three regimes: A locked regime where both fronts travel at the same speed, a pulled regime where the secondary front travels at the linear spreading speed associated to the intermediate state, and an accelerated regime where the selected speed is between these two speeds. We show that the transition to locked fronts can be described by the crossing of a resonance pole in the linearization about the primary front. In addition, using properties of this resonance pole we derive the selected wavespeed in the accelerated case and dete...

Attractor reconstruction from interspike intervals is incomplete

Abstract We investigate the problem of attractor reconstruction from interspike times produced by an integrate-and-fire (IF) model of neuronal activity. Suzuki et al. [Biol. Cybernet. 82 (2000) 305–311] found that the reconstruction of the Rossler attractor is incomplete if the IF model is used. We explain this failure using two observations. One is that the attractor reconstruction only reconstructs an attractor of a discrete system, which may be a strict subset of the original attractor. The second observation is that for a set of parameters with non-empty interior numerical simulations demonstrate that the attractor of the discrete system is indeed a strict subset of the Rossler attractor. This is explained by the existence of phase locking in a nearby periodically forced leaky IF model.

Existence and Stability of Traveling Pulses in a Reaction-Diffusion-Mechanics System

In this article, we analyze traveling waves in a reaction–diffusion-mechanics (RDM) system. The system consists of a modified FitzHugh–Nagumo equation, also known as the Aliev–Panfilov model, coupled bidirectionally with an elasticity equation for a deformable medium. In one direction, contraction and expansion of the elastic medium decreases and increases, respectively, the ionic currents and also alters the diffusivity. In the other direction, the dynamics of the R–D components directly influence the deformation of the medium. We demonstrate the existence of traveling waves on the real line using geometric singular perturbation theory. We also establish the linear stability of these traveling waves using the theory of exponential dichotomies.

Linear spreading speeds from nonlinear resonant interaction

We identify a new mechanism for propagation into unstable states in spatially extended systems, that is based on resonant interaction in the leading edge of invasion fronts. Such resonant invasion speeds can be determined solely based on the complex linear dispersion relation at the unstable equilibrium, but rely on the presence of a nonlinear term that facilitates the resonant coupling. We prove that these resonant speeds give the correct invasion speed in a simple example, we show that fronts with speeds slower than the resonant speed are unstable, and corroborate our speed criterion numerically in a variety of model equations, including a nonlocal scalar neural field model.

Wavetrain solutions of a reaction-diffusion-advection model of mussel-algae interaction

We consider a system of coupled partial differential equations modeling the interaction of mussels and algae in advective environments. A key parameter in the equations is the ratio of the diffusion rate of the mussel species and the advection rate of the algal concentration. When advection dominates diffusion, one observes large-amplitude solutions representing bands of mussels propagating slowly in the upstream direction. Here, we prove the existence of a family of such periodic wavetrain solutions. Our proof relies on geometric singular perturbation theory to construct these solutions as periodic orbits of the associated traveling wave equations in the large-advection--small-diffusion limit. The construction encounters a number of mathematical obstacles which necessitate a compactification of phase space, geometric desingularization to remedy a loss of normal hyperbolicity, and the application of a generalized exchange lemma at a loss-of-stability turning point. In particular, our analysis uncovers log...

Estimating epidemic arrival times using linear spreading theory

We study the dynamics of a spatially structured model of worldwide epidemics and formulate predictions for arrival times of the disease at any city in the network. The model is composed of a system of ordinary differential equations describing a meta-population susceptible-infected-recovered compartmental model defined on a network where each node represents a city and the edges represent the flight paths connecting cities. Making use of the linear determinacy of the system, we consider spreading speeds and arrival times in the system linearized about the unstable disease free state and compare these to arrival times in the nonlinear system. Two predictions are presented. The first is based upon expansion of the heat kernel for the linearized system. The second assumes that the dominant transmission pathway between any two cities can be approximated by a one dimensional lattice or a homogeneous tree and gives a uniform prediction for arrival times independent of the specific network features. We test these predictions on a real network describing worldwide airline traffic.